Association schemes of quadratic forms and symmetric bilinear forms

Wang, Yangxian ; Wang, Chunsen ; Ma, Changli ; Ma, Jianmin

Journal of Algebraic Combinatorics, Tome 17 (2003) no. 2, pp. 149-161.

Résumé

Summary: Let $X _{n}$ and $Y _{n}$ be the sets of quadratic forms and symmetric bilinear forms on an $n$-dimensional vector space $V$ over $\mathbb F _{ q}$ mathbbF_q , respectively. The orbits of $GL _{n}( \mathbb F _{ q}$ mathbbF_q ) on $X _{n} \times X _{n}$ define an association scheme $Qua( n, q)$. The orbits of $GL _{n}( \mathbb F _{ q}$ mathbbF_q ) on $Y _{n} \times Y _{n}$ also define an association scheme $Sym( n, q)$. Our main results are: $Qua( n, q)$ and $Sym( n, q)$ are formally dual. When $q$ is odd, $Qua( n, q)$ and $Sym( n, q)$ are isomorphic; $Qua( n, q)$ and $Sym( n, q)$ are primitive and self-dual. Next we assume that $q$ is even. $Qua( n, q)$ is imprimitive; when $( n, q)ne$ (2,2), all subschemes of $Qua( n, q)$ are trivial, i.e., of class one, and the quotient scheme is isomorphic to $Alt( n, q)$, the association scheme of alternating forms on $V$. The dual statements hold for $Sym( n, q)$.

Détail

Keywords: association scheme, quadratic form, symmetric bilinear form